When, then, is \(\mathfrak {a}_{\mathrm {vec},F}\)  “small”, i.e., equal to \(\aleph _1\) ? It is shown in [1]} that \(\mathfrak {a}_{\mathrm {vec},F}\)  is small in the Cohen and (iterated or side-by-side) Sacks models. It was conjectured in [1]} that this is also the case in the Miller model (described at the end of § below), and moreover, that a certain parametrized \(\diamondsuit \)  principle, in the sense of Moore, Hrušák, and Džamonja [3]}, suffices. It is this conjecture that we verify in Theorem REF  below.The conjecture in [1]} states “\(\diamondsuit (\mathbb {N}^\mathbb {N},=^\infty )\)  implies \(\mathfrak {a}_{\mathrm {vec},F}=\aleph _1\) ”; this is, possibly, slightly stronger than what we prove here, cf. Lemma REF . As a consequence of the main result from [3]} (Theorem REF  below), in essentiall all “canonical” models of set theory, i.e., those obtained by \(\omega _2\) -length countable support iterations of definable proper forcings, \(\mathfrak {a}_{\mathrm {vec},F}=\mathrm {non}(\mathcal {M})\) .
